Using the neem oil with a phosphate free dish soap is fine, you just need a couple drops though. Just enough to lower the surface tension so the oil spreads evenly over the leaves. Don't forget to coat the top and bottom of all the leaves, especially if you're using neem oil as it'll act as a horticultural oil by suffucating insects/eggs. Azadirachtin acts as a natural systemic insecticide, it's absorbed by the leaves then will poison anything feeding on them disrupting their hormone levels. It's not effective against all insects though, so proper identification is important. The difference between neem oil and azamax is azamax solely uses azadirachtin to control the infestation, whereas neem oil uses both the chemical and suffocating properties of hort. oils. The difference between neem oil/azamax and gnatrol is that gnatrol is a bacterial innoculant meant for fungus gnats, it's like unleashing the black plague on them. If you only have a couple though, you're better off letting the pots dry out more between waterings like you've already found out.
They suggest adding Pro-Tekt because it's a potassium/silicon additive, both minerals are needed to build thick cell walls which help prevent insect/fungal/viral problems. You're better off investing in a better environment to grow in first though, better lighting and air flow will make a bigger difference then nutrient additives. Once the environment is dialed in, then it's worth investing in supplements to increase growth. Until then you get more bang for your buck elsewhere.
